Generalized little \(q\)-Jacobi polynomials as eigensolutions of higher-order \(q\)-difference operators (Q2701623)

The authors consider the polynomials \(p_n(x;a,b;M)\) which are orthogonal with respect to the weight function for the little \(q\)-Jacobi polynomials together with a discrete mass \(M\) at \(x=0\). It is shown that for \(a=q^j\) with \(j=0,1,2,\ldots\) these polynomials are eigenfunctions of a linear \(q\)-difference operator of order \(2j+4\) with polynomial coefficients. The method of constructing the linear \(q\)-difference operator was developed by the second author in [\textit{A.~Zhedanov}, J. Comput. Appl. Math. 107, No. 1, 1-20 (1999; Zbl 0929.33008)] and is based on the use of the so-called Geronimus transformation. This method does not work in the general case where the parameters satisfy \(0<aq<1\) and \(b<q^{-1}\), but does in the special case that \(a=q^j\) with \(j=0,1,2,\ldots\). Since the little \(q\)-Jacobi polynomials reduce to the little \(q\)-Laguerre polynomials for \(b=0\), a similar result is derived for generalized little \(q\)-Laguerre polynomials obtained by adding a mass \(M\) at \(x=0\) to the orthogonality measure for the little \(q\)-Laguerre polynomials. These results are partly \(q\)-analogues of much more general results obtained in [\textit{J.~Koekoek} and \textit{R.~Koekoek}, Proc. Am. Math. Soc. 112, No. 4, 1045-1054 (1991; Zbl 0737.33003); J. Comput. Appl. Math 126, No. 1-2, 1-31 (2000; Zbl 0970.33004)].